experimental verification of folded glassthe folded plates. a frame work was made where the load...

Experimental Verification of Folded Glass

Plates Used in Architectural Window

Glazing

C.V. Girija Vallabhan', M.M. El-Shami',

K.S. Kandil*& O.M. Tawfik

* Texas Tech University, Lubbock, Texas, USA

Menoufia University, Menoufia, Egypt.

Introduction

Architects use large glass panels as vertical folded plate structures inbuildings for allowing lights as well as aesthetic appearances. Often thesefolded glass plate structures are constructed using large plane glass platesplaced vertically, but each of them are connected at an angle 90 degrees ormore. These panels are subjected to lateral wind loads and are glued togetherat their joints with a silicone sealant. Till now, to the knowledge of theauthors, no report has been published on the analysis of these folded glassplate structures. These structures are generally supported at the top and thebottom within an aluminum frame with rubber gaskets to release stressconcentrations that might be resulting from some small constructionirregularities. Hence they can be assumed to be simply supported on all theedges.

A model experiment is made to determine the displacements andstresses within the structure. Thin Aluminum plates are used instead of glassplates, and these plates are glued together using silicone sealant. In order tosimulate uniform pressure loading on the structural system, an enclosedchamber is made and the air inside is sucked out. Displacement transducersand strain gages are mounted on the system to measure the displacements andstrains as the pressure is reduced inside the chamber. Later, an exactly similarsetup is made where the joints are welded together so that the moments aretransferred from one plate to the other. Results are obtained for both cases

A nonlinear higher order finite element program using 9 nodedquadrilateral elements is made specifically for the analysis of this structural

Transactions on Modelling and Simulation vol 16, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X

26 Computer Methods and Experimental Measurements

system. Results obtained from the experiments are compared with finiteelement solutions.

Stress Analysis of Folded Glass Plates

When the lateral displacement of flat window glass plates subjected touniform pressures becomes sufficiently large compared its thickness, theirbehavior is highly nonlinear even though elastic. Glass plates have a suddenbrittle fracture when they break. Using von Karman nonlinear theory of plates^and finite difference method, Vallabhan* analyzed the displacement andstresses in rectangular glass windows. Beason* and Vallabhan and Minof andmany other researchers have performed detailed experiments and shown thatvon Karman equations are good for analysis of window glass plates. It becameapparent to the authors that analysis of folded glass plates have to includegeometrical nonlinearity.

Folded plate structures are shell structures which gain their stiffness byvirtue of their efficient membrane action. The obvious choice for the analysisis to use nonlinear shell finite elements. In finite element analysis of plates,researchers* have found that Mindlin theory of plates^ which includes theshear deformation in the plates, is convenient than the classical Kirchhofftheory. Here, a nine-noded quadrilateral finite element is developed and usingthis model, El Shami, et al/ have shown that solutions for displacements andstresses using nonlinear Mindlin and von Karman theories, are almost thesame for thin plates subjected to lateral pressures.

Nonlinear Quadrilateral Mindlin Shell Element

Since the general theory and solution techniques are published earliermany times (Zienkiewicz ), only very brief account of the theory is presentedhere. While including the shear deformations in the plate, Mindlin used the ideathat the final rotations of the normal to the plate are obtained by adding therespective derivatives of the lateral displacement function w(x,y) with respect tox and y with the corresponding vertical shear deformations. All otherassumptions in his theory are the same as those used in the von Karman theory.Thus, the displacements at any point (x,y,z) are

v(x,y,z) = vx,y-z0;x,y (1)

w(x,y,z) = w(x,y)

where u(x,y),v(x,y) and w(x,y) are the displacements of the middle surface

and 0; , 0y are the rotations of the normal in the undeformed plate in the 'jyz'


Computer Methods and Experimental Measurements 27

and 'zx' planes respectively after deformation. For a Mindlin plate the relevantnonlinear strain vector is given as

. - r, r.

where the linear midplane strains are the membrane strain, is

the bending strain is {e * } ' = {<%,, - 0^

the shear strain is |fi£ | ' = 1 6^ + w ., - 0, + w J ,

and finally the nonlinear component of membrane strain is

In the finite element model, the displacement functions and rotations areinterpolated using the Lagrangian shape functions for a 9 noded quadrilateralelement. The equations are

where ^.^,17, i = 1,9 are the shape functions of 9-noded quadrilateral

element. ,77 are coordinates that are used to transform the quadrilateral into asquare. The nodal displacement vector

{%}' ={%, V, w, , ^ ^ . .}% (4)

In other words, this vector has 5 d.o.f. per node and with 9 nodes, there are 45d.o.f. per element. It should be noted that an additional degree of freedom

representing the rotation about the z-axis, i.e., #, has to be included for thegeneral shell element analysis. Using Eq.(2), and (3), the relationshipsbetween the three parts of the strain and nodal displacements can be written as

Once the strain displacement equations are established, the element stiffnessmatrix in incremental form is written as

(6)



where [^J is tangent stiffness matrix, JAwj is nodal displacement increment,

and [ AjpJ is nodal force increment. The formulation presented here is referred

to as 'total Lagrangian' and is given in reference*. The tangent stiffness matrixcan be written as

and \Kj\ is defined as the geometric or "initial stress" stiffness matrix '. At

this point, the additional degree of freedom 0, is added to the displacementvector, making the total degrees of freedom as 54 per shell element.

The computation of the element stiffness matrices requires reducedintegration technique for the part of the stiffness due to vertical shear. Theforces during the equilibrium check are calculated using the equation,

(8)V

A Newton-Raphson* iterative procedure is used in this model forsolving the nonlinear equations, where the unbalanced force equation is

{*}* (9)

for the A-th iteration and all the incremental displacements are added together

until, the unbalanced force vector is negligible or the norm of the {Aw}* issmall. Several problems were solved and the results are compared to classicalsolutions by Levy* and finite difference solutions by Vallabhan*. Goodagreement was observed between the solutions.

Finite Element Model to Include Hinged Joints between Plates

The finite element model so far discussed will deform with a singlevalue of joint rotation. But in the folded glass curtain wall system, the platesare connected together at the joints with silicone. This connection will notallow moments to be transferred and will produce multiple values of rotationsat the hinged joints with a single value for the joint displacement vector. Toincorporate multiple values of joint rotations along the longitudinal axis, thefinite element model was modified, by introducing two rotations along that



axis for both adjoining elements, at the same time all other joint displacementand rotation parameters were kept uniquely compatible. Problems were solvedusing this model, and some of the results were compared with simple exactsolutions and results matched very well.

Experiments on Folded Plates

Three different tests were made, and in all of them, aluminum plateswere used instead of glass plates. Aluminum has almost same the modulus ofelasticity as glass plate. The assembly contains 8 panels of equal size, and itscross section is shown in Fig. 1. The overall dimensions of the entireassembly were 48 in. (1.2192 m) by 26.87 in. (0.6825 m). Each 8 panels were48 in. (1.2192 m.) X 4.75 in., (0.12065 m.) with a thickness equal to 1/8 in.(3.175 mm). The modulus of elasticity of the aluminum plate was tested to be10.348 X 10* psi (71.4 GPa). Three displacement transducers and three straingauges were used to measure the displacements and the strains in differentlocations. The results were collected through a data acquisition system. Thethree different experimental setups were employed and they are described inthe following.

1- The first test set up was made to simulate a folded plate glass curtainwall. See Fig. 2. The aluminum plates were used instead of glass platesand were connected together at their joints using silicone. In order tosimulate a uniform pressure on the folded plates, the entire setup wascovered in a welded steel chamber and a suction pressure was appliedinside the chamber. The suction pressure indirectly applied a uniformlateral pressure on the aluminum plates. The vacuum pressure machinehad a capacity to apply 10 psi. (69 kPa) . Displacements were measured atvarious points, but a sudden failure occurred in the folded glass platesystem at about 1.18 psi. (8.16 kPa) pressure. The failure occurred in thesilicone sealant. As the pressure increased, the silicone sealant failed inshear at the joints at the bottom joints where silicone seals were exposedto the vacuum pressure. The load-displacement relations are shown in Fig.3. The finite element model developed in this research was used to analyzethe load-displacement pattern and the results are shown in Fig 4. It wasinteresting to note that the results compared very well. Even though, it wasexpected that the silicone sealant in the system would fail at first, we didnot expect it to fail that fast.

2- Because of the premature failure in the first setup, it was decided to weldthe aluminum plates together to study its load-deformation characteristics.Here the load was applied to a vacuum pressure equal to 6.5 psi. (44.85



kPa.) and the load-displacement relations both in the experiment and thefinite element model were linear. The relations are also shown in Fig. 4.

3- Another set up was fabricated, so that external load could be applied onthe folded plates. A frame work was made where the load could be appliedthrough a previously calibrated hydraulic jack which measures thepressure applied through the jack. In this case also, the aluminum plateswere welded together as in the setup 2. In this case, the boundaryconditions were made such that the folded plates were connected to theside diaphragms with silicone along its shorter sides, while thelongitudinal edges were free to displace outwards. This setup is shown inFig. 5. This condition was purposely setup to see whether the folded platesmight buckle under loads to the sides. In this test, the magnitude of theapplied concentrated load was 4,233 pounds(18.77 kN) and some portionof the folded plate began to yield with plastic deformations. In this setup,the displacements were measured at % span instead of the center as wasdone in the first two setups. The load-displacement relations for this setupis shown in Fig. 6, and the results are compared with finite elementsolutions.

Conclusions

The research was directly aimed at determining the strength of foldedglass plate structures. The following conclusions are reached from thisresearch.

1. The weakest link in the folded plate aluminum structure connectedtogether by silicone sealant, is the sealant itself. The system fails inshear as the uniform pressure was applied on the whole system.

2. There is very little difference between the load-displacementcharacteristics of the system which is connected together bysealant or welded together. The major difference is that, the oneconnected together by sealant cannot take much of the pressure, asthe weakest link is the sealant.

3. There is very good shell action in the structure, and for the loadsapplied, there is almost negligible geometric nonlinearity.

4. The load-displacement analysis of the system is linear, and hence alinear finite element model is sufficient.



5. Material nonlinearity may be more predominant, in the case ofaluminum folded plates, than geometric nonlinearity. But thematerial nonlinearity is not applicable to glass plates, as glass failswith brittle fracture.

6. The finite element model developed can be used to analyzedisplacements and stresses in folded glass plate curtain wallssubjected to lateral pressures.

Acknowledgments

This work is based on research accomplished in the Civil EngineeringDepartment at Texas Tech University. The research is supported by theEgyptian Government through the channel system programs between TexasTech University and El- Menoufia University.

References

1. Beason, L., "A Failure Prediction Model for Window Glass", Ph.D.,Dissertation, Texas Tech University, Lubbock, Texas, 1980.

2. Dym, C. L., and Shames, I. H., "Solid Mechanics A VariationalApproach," McGraw-Hill Book Co., N. Y., 1973.

3. El Shami, M.M., Vallabhan, C.V.G. and Kandil, K., "Comparison ofNonlinear von Karman and Mindlin Plate Solutions", Proc. Of TexasASCE Conf., Houston, April,6-8, 1997.

4. Levy, S., "Square Plate with Clamped Edges Under Normal PressureProducing Large Deflections," NACA, Report No. 740, 1942.

5. MacNeal, R. H.,"A Simple Quadrilateral Shell Element," Computers andStructures, Vol. 8, pp. 175-183., 1978.

6. Vallabhan, C. V. G., "Iterative Analysis of Nonlinear Glass Plates,"Journal of the Structural Division, Proceedings of the American Society ofCivil Engineers, Vol. 109, ST2, pp. 489-502, February, 1983.

7. Vallabhan, C. V. G., and Minor, J. E., "Experimentally VerifiedTheoretical Analysis of Thin Glass Plates," Computational Methods andExperimental Methods, July, 1984.

8. Weaver, W. J., and Johnston, P. R., "Finite Elements for StructuralAnalysis," Prentice-Hall, Inc. Englewood Cliffs, N. J., 1984.

9. Zienkiewicz, O. C., and Taylor, R.L., "The Finite Element Method,"McGraw-Hill Book Co., London, 1994.


32

to Vacuum!

Computer Methods and Experimental Measurements

0.171m

Fig. 1. Cross Section of the Folded Plate and Chamber

Fig. 2. A Photograph of the Experimental Setup



Fig. 3. Failure of the Folded Plate in the Silicone

0.5

A Exper Silicone_ _Q _ Theory Silicone I_o -ExperWelded_ _ _ _ _ Theory Welded

10 20 30 40 50Vacuum (kPa)

Fig. 4. Load Displacement Relationships at the Center of the Setup(Test # 1 and Test # 2)



Fig. 5. Experimental Setup to Apply Concentrated Load

25

2

1 5

11

05

0 ;(

r*

) 1000 20

I j *

_g^

V^^ '

00 3000 4000 50

Load (ib)

+ BcperB— Theory

00

Fig. 6. Load Displacement relationships at the Center of Cross Section atSpan (Test # 3)


experimental verification of folded glassthe folded plates. a frame work was made where the load...

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